A Simple Continuous Mixture Droplet Evaporation Model with Multiple Distribution Functions
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1 Introduction A Simple Continuous Mixture Droplet Evaporation Model with Multiple Distribution Functions William Hallett and Claire Grimwood Dept. of Mechanical Engineering, University of Ottawa, Ottawa, Canada K1N 6N5 hallett@genie.uottawa.ca Continuous thermodynamics, which represents a mixture as a continuous probability density function rather than as discrete components, has proved an effective means of modelling complex multicomponent mixtures such as commercial liquid fuels. The technique has been applied to droplet evaporation [1,2] and used to predict fuel vapour composition variations in a Diesel engine cylinder [3]. However, so far but a single distribution function has been used, representing the n- paraffin family of compounds. Real petroleum fuels, on the other hand, contain several groups of hydrocarbons: paraffins, aromatics, naphthenes, and sometimes discrete components such as fuel additives as well. To model these different groups, a separate distribution function is required for each group as well as for each discrete species, if present. This paper presents a model for doing this which is simple enough to be suitable for inclusion in a spray combustion code. It is an extension of classical quasi-steady droplet evaporation theory, and makes the same simplifying assumptions: gas phase quasi-steadiness, spherical symmetry, and constant properties evaluated at a suitable reference state. The liquid phase is assumed to be well-mixed, so that the droplet temperature and composition are uniform, an assumption justified elsewhere in these proceedings [4]. Vapour-Phase Transport Equations The mixture is assumed to contain J different families of hydrocarbons, each represented by a vapour phase distribution function f j (I) with mean j and variance j2, and a liquid phase distribution f Lj (I) with parameters Lj and Lj 2 The distribution variable I can be any convenient property; here we choose the species molecular weight. The vapour and liquid phase mol fractions of a particular species are given by where y Fj and x Fj are the overall mol fractions of distribution j in the vapour and liquid phases. Assuming a quasi-steady vapour phase permits the full transport equations given elsewhere in these proceedings [4] to be simplified to: (1) (2) (3) where is a mean diffusivity as defined in [1], and j represents either (y Fj j ) or (y Fj j ), where = is the second central moment. The boundary conditions are values of y Fj, (y Fj j ), and
2 (y Fj j ) at the surface and in the ambient. The evaporating mol flux N is related to the molar velocity v * by continuity: Equations (2) and (4) may be solved for N: (4) (5) where j is the ratio of the mol flux N j of distribution j to the total flux N, (6) The j 's may be found by writing eq. (5) for j = 1 and for j and equating, giving (7) From eq. (6), we have the requirement that (8) These two expressions give J equations in J unknowns which can be solved for the j. The distribution numbered j = 1 should be the one with the highest mol fraction in the vapour phase. For the special case in which all the 's are equal, the solution simplifies to j = y FjR / y FR, similar to the result given by Law and Law [5] for discrete components. Equations (2) and (3) can be solved for the variation of y Fj and in the vapour phase: (9) (10) where Z = R / r. The energy equation and its solution, required to calculate heat transfer to the droplet, are as given in [2], except that the fuel specific heat is now the mol fraction weighted sum of those for the individual distributions. Other features of the model are as in [2]. Liquid Phase Balances The change in composition of the liquid phase as evaporation proceeds is determined by flux balances at the droplet surface. Extending the expressions for a well-mixed droplet given in [1] to multiple distributions yields (11)
3 (12) with a similar expression for L. As in earlier work [1,2], phase equilibrium is described by a continuous mixture version of Raoult's Law and the Clausius-Clapeyron equation, and gamma distribution functions were used for both liquid and vapour phases. The enthalpy of vaporization equation in [4] simplifies to (13) In applying this model to a spray code, the composition of the vapour flux leaving the surface is required for the vapour source term in the gas phase computations. It is given by distribution functions f Sj describing the component mol fluxes. For the quasi-steady model, the expressions in [4] reduce to (14) which if y Fj = 0 reduces to Sj = jr. Distribution Functions and Properties Table I: Fuel Specifications Gasoline Diesel paraffins (mass/mol %) 70.0/ /69.2 benzenes 30.0/ /21.4 naphthalenes /9.4 ASTM 10% ( C) ASTM 90% Table II: Distributions for Fuel Fractions Group L0 L0 n-paraffins (for gasoline) n-paraffins (for Diesel) alkylbenzenes naphthalenes To illustrate the capabilities of the theory, sample calculations are presented for a gasoline and a Diesel fuel, whose compositions and ASTM D-86 distillation properties are given in Table I. The gasoline was assumed composed of n-paraffin and alkylbenzene fractions, while the Diesel fuel included alkylnaphthalenes as well. Distribution functions for these fractions are given in Table II, and were selected as follows. The two aromatic groups were first assigned distribution origins corresponding to the lowest molecular weight members (benzene and naphthalene) and upper bounds were established arbitrarily by setting the 99% point on the cumulative distribution at fully methylated benzene and naphthalene respectively (C 12 and C 16 ). Within these constraints, means and standard deviations were selected to give reasonably symmetrical distributions. The n-paraffin fraction parameters were then fitted so that the 10% and 90% distillation points for the fuel as a whole matched those in Table I. A continuous mixture simulation of the ASTM D-86 test was used for this, and the origin set to = 44 (= propane). The distribution functions for these fuels are graphed in Figs. 1 and 2. Correlations for properties are presented in the Appendix.
4 Fig. 1: Distribution functions for simulation of gasoline. Fig. 2: Distribution functions for D iesel fuel. Results Sample calculations are presented for a 100 m droplet of initial temperature 300K, vaporizing in surroundings at 1000K. Figs. 3-5 show predictions for the gasoline droplet. As lighter components are distilled out of the mixture, the distribution means increase and the standard deviations drop (Fig. 3); this causes the bubble point temperature to rise steadily, and after an initial droplet heating transient the liquid temperature parallels the bubble point (Fig. 4). (It has been shown [2] that the parallel behaviour of liquid and bubble point temperatures indicates a quasi-equilibrium state of vaporization.) As the paraffin fraction is the lighter of the two, almost all of the vapour produced initially is paraffin ( PAR, Fig. 3), and consequently only the paraffin fraction composition is affected by the loss of light components. However, somewhat later this leads to the mean molecular weight of the paraffins crossing over that of the aromatic fraction, and from henceforth the vapour produced is increasingly aromatic in character. The distribution parameters for the vapour mol flux indicate the changes in the vapour composition produced over time (Fig. 5), and again the paraffins begin changing right at the start while the aromatics are not affected until much later. The standard deviation for the paraffin vapour rises initially as droplet heating increases the vapour pressures of heavier species, but drops later in response to the loss of light components. Figs. 6-8 show predictions for the Diesel fuel. Here the alkylbenzenes are the lightest fraction, but they also have a much narrower distribution than the paraffins; consequently their parameters change earlier than the paraffins, but the changes are not so great (Fig. 6). They are fairly quickly distilled out of the droplet (Fig. 7), but form a disproportionately large part of the early vapour produced (Fig. 8). The naphthalenes are the next to be eliminated, leaving the heaviest paraffins as the last fraction before the droplet disappears. The vapour flux distributions show that the vapour produced is continually changing in character and composition as evaporation proceeds. Conclusions The model presented here is simple enough to be incorporated in a spray code, yet has the potential to simulate all the composition changes that take place in commercial fuel droplets and their vapour as evaporation proceeds. The model can easily be extended to include other hydrocarbon groups; all that is required are additional property correlations for them.
5 Fig. 3: Liquid-phase distribution means and standard deviations for gasoline. Fig. 4: Bubble point, liquid temperature and mass % evaporated; paraffin mol fraction in vap our flux and in liquid for gasoline. Fig. 5: Distribution parameters of the vapour mol flux for gasoline. Fig. 6: Liquid-phase distribution means and standard deviations for Diesel. Fig. 7: Liquid temperature, mass % evaporated, and liquid compone nt mol fractions for D iesel. Fig. 8: Distribution parameters of the vapour mol flux for Diese l.
6 Acknowledgement The authors are grateful to NSERC for financial support of this work. Nomenclature c molar density, kmol/m 3 D diffusivity, m 2 /s f(i) distribution function I distribution variable, = mol mass, kg/kmol N mol flux, km ol/m 2 s R droplet radius, m v 8 molar average velocity, m/s y mol fraction References distribution origin distribution mean (= mean mol mass) j ratio N j / N dist'n standard deviation = (2nd central mo ment) Subscripts: ARO alkylbenzenes B bubble point F fuel j distribution function index L liquid phase NAP naphthalenes PAR n-paraffins R at droplet surface S vapour mol flux ( source ) 0 initial value ambient value [1] Tamim, J., H allett, W.L.H., Chem. Eng. Sci. 50, 2933 (1995). [2] Hallett, W.L.H., Comb. & Flame 121, 334 (2000). [3] Lipp ert, A.M., Stanton, D.W., Rutland, C.J., Hallett, W.L.H., Reitz, R.D., Int. J. Engine Research 1, 1, [4] Abdel-Qader, Z., Hallett, W.L.H., elsewhere in these proceedings. [5] Law, C.K., Law, H.K., AIAA J. 20, 522 (1982) [6] Cho u, G.F., P rausnitz, J.M., Fluid P hase Equilibria 30, 75 (1986). Appendix: Properties Correlations for Alkylbenzenes and Alkylnaphthalenes Correlations for n-paraffin properties a s functions o f molecula r weight I and temperature are taken from [1], except that for the gasoline simulation a boiling point correlation more suited to lower molecular weight paraffins was used (T B = I). New correlations were developed for aromatics; they have the same form as in [1], so that only the values of the parameters are given here. The vapour specific heat was taken from the correlations of Chou and Prausnitz [6]. Correction: The value for b KT in [1] should read rather than Coefficient Alkylbenzenes Napthalenes Boiling point a B b B Diffusivity a D 3.551E E-09 b D E E-12 b Critical temperature a CR b CR Critical pressure ( = a P + b P I, units bar) a P b P Coefficient Alkylbenzenes Napthalenes Therm al conductivity a KT 1.086E E-05 b KT E E-07 a KC E E-02 b KC 2.220E E-05 Enthalpy of vaporization a H 8.286E E+07 b H 2.606E E+04 (T CR - T B ) Liquid specific heat a L b L 1.51E E-03 c L 2.75E E-06
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